Weil--Petersson geodesics on the modular surface

Gadre, V. (2020) Weil--Petersson geodesics on the modular surface. arXiv, (Unpublished)

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Publisher's URL: https://arxiv.org/abs/2011.12034

Abstract

We consider the Weil–Petersson (WP) metric on the modular surface. We lift WP geodesics to the universal cover of the modular surface and analyse geometric properties of the lifts as paths in the hyperbolic metric on the universal cover. For any pair of distinct points in the thick part of the universal cover, we prove that the WP and hyperbolic geodesic segments that connect the pair, fellow-travel in the thick part and all deviations between these segments arise during cusp excursions. Furthermore, we give a quantitative analysis of the deviation during an excursion. We leverage the fellow traveling to derive a correspondence between recurrent WP and hyperbolic geodesic rays from a base-point. We show that the correspondence can be promoted to a homeomorphism on the circle of directions. By analysing cuspidal winding of a typical WP geodesic ray, we show that the homeomorphism pushes forward a Lebesgue measure on the circle to a singular measure. In terms of continued fraction coefficients, the singularity boils down to a comparison that we prove, namely, the average coefficient is bounded along a typical WP ray but unbounded along a typical hyperbolic ray.

Item Type:Articles
Status:Unpublished
Refereed:No
Glasgow Author(s) Enlighten ID:Gadre, Dr Vaibhav
Authors: Gadre, V.
Subjects:Q Science > QA Mathematics
College/School:College of Science and Engineering > School of Mathematics and Statistics > Mathematics
Journal Name:arXiv
Copyright Holders:Copyright © 2020 The Author
Publisher Policy:Reproduced under a Creative Commons License

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